Question

Consider the following data for a dependent variable y and two independent variables, x2 and x1.

x1 x2 y

30 12 94

46 10 109

24 17 113

50 17 179

41 5 94

51 19 175

75 8 171

36 12 118

59 13 143

77 17 212

Round your all answers to two decimal places. Enter negative values as negative numbers, if necessary.

a. Develop an estimated regression equation relating y to x1 . Predict y if x1=45.

b. Develop an estimated regression equation relating y to x2. Predict y if x2=25.

c. Develop an estimated regression equation relating y to x1 and x2. Predict y if x1=45 and x2=25.

Answer 1

Part a)  Estimated regression equation relating y to x1

The following data are passed:

x1	y
30	94
46	109
24	113
50	179
41	94
51	175
75	171
36	118
59	143
77	212

The independent variable is x1, and the dependent variable is y. In order to compute the regression coefficients, the following table needs to be used:

	x1	y	x1*y	x12	y2
	30	94	2820	900	8836
	46	109	5014	2116	11881
	24	113	2712	576	12769
	50	179	8950	2500	32041
	41	94	3854	1681	8836
	51	175	8925	2601	30625
	75	171	12825	5625	29241
	36	118	4248	1296	13924
	59	143	8437	3481	20449
	77	212	16324	5929	44944
Sum =	489	1408	74109	26705	213546

Based on the above table, the following is calculated:Therefore, based on the above calculations, the regression coefficients (the slope m, and the y-intercept n) are obtained as follows:

Therefore, we find that the regression equation is:

y=48.7428+1.8826x1

Predict y if x1=45

y=48.7428+1.8826*(45)

= 133.4598 answer

Part b estimated regression equation relating y to x2.

The following data are passed:

x2	y
12	94
10	109
17	113
17	179
5	94
19	175
8	171
12	118
13	143
17	212

The independent variable is x2, and the dependent variable is y. In order to compute the regression coefficients, the following table needs to be used:

	x2	y	x2*y	x22	y2
	12	94	1128	144	8836
	10	109	1090	100	11881
	17	113	1921	289	12769
	17	179	3043	289	32041
	5	94	470	25	8836
	19	175	3325	361	30625
	8	171	1368	64	29241
	12	118	1416	144	13924
	13	143	1859	169	20449
	17	212	3604	289	44944
Sum =	130	1408	19224	1874	213546

Based on the above table, the following is calculated:

Therefore, based on the above calculations, the regression coefficients (the slope m, and the y-intercept n) are obtained as follows:

Therefore, we find that the regression equation is:

y=75.8+5x2

Predict y if x2=25

y=75.8+5*(25)

= 200.8

Part c estimated regression equation relating y to x1 and x2.

These are the data that have been provided for the dependent and independent variables:

yy	x1x1	x2x2
94	30	12
109	46	10
113	24	17
179	50	17
94	41	5
175	51	19
171	75	8
118	36	12
143	59	13
212	77	17

The following matrix and vector are defined in order to conduct the matrix calculation required to compute the estimated multiple regression coefficients:

Now, the vector with the estimated regression coefficients β​ is computed through the following matrix operation:

β​=(X′X)⁻¹X′y

Therefore, based on the data provided, the estimated multiple linear regression equation is:

y=−15.5982+1.8772x1 + 4.9694x2

Predict y if x1=45 and x2=25.

y=−15.5982+1.8772*(45) + 4.9694*(25)

= 193.1108

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